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Kruskal Algorithm

Kruskal's algorithm is a minimum spanning tree algorithm that finds the minimum cost collection of edges that connect all vertices in a connected, undirected graph. It works by sorting the edges by weight and then sequentially adding the shortest edge that does not create a cycle. The algorithm uses a disjoint set data structure to track connected components in the graph. It runs in O(ElogE) time, where E is the number of edges.

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KRUSKAL’S ALGORITHM MADE BY :- DHRUV PATEL(160110116029). BHAVIK VASHI(160110116061).
Introduction • Kruskal's algorithm is a minimum-spanning-tree algorithm which finds an edge of the least possible weight that connects any two trees in the forest. • Some key terms:- 1. spanning tree:- A spanning tree of a connected, undirected graph G is a sub-graph of G which is a tree that connects all the vertices together. 2. forest:- A forest is a disjoint union of trees, or equivalently an acyclic graph that is not necessarily connected.
Kruskal algorithm EXAMPLE:- A cable company want to connect five villages to their network which currently extends to the market town of Avonford. What is the minimum length of cable needed?
A F B C D E 2 7 4 5 8 6 4 5 3 8 List the edges in order of size: ED 2 AB 3 AE 4 CD 4 BC 5 EF 5 CF 6 AF 7 BF 8 CF 8
Select the shortest edge in the network ED 2 A F B C D E 2 7 4 5 8 6 4 5 3 8
Select the next shortest edge which does not create a cycle ED 2 AB 3 CD 4 AE 4 A F B C D E 2 7 4 5 8 6 4 5 3 8
All vertices have been connected. The solution is ED 2 AB 3 CD 4 AE 4 EF 5 Total weight of tree: 18 A F B C D E 2 7 4 5 8 6 4 5 3 8 Solution
8 1. A ←  2. for each vertex v  V 3. do MAKE-SET(v) 4. sort E into non-decreasing order by w 5. for each (u, v) taken from the sorted list 6. do if FIND-SET(u)  FIND-SET(v) 7. then A ← A  {(u, v)} 8. UNION(u, v) 9. return A Running time: O(V+ElogE+ElogV)=O(ElogE) – dependent on the implementation of the disjoint-set data structure KRUSKAL(V, E) O(V) O(ElogE) O(E) O(logV)
Disjoint data structure • MAKE-SET(u) – creates a new set whose only member is u • FIND-SET(u) – returns a representative element from the set that contains u • UNION(u, v) – unites the dynamic sets that contain u and v, say Su and Sv • E.g.: Su = {r, s, t, u}, Sv = {v, x, y} UNION (u, v) = {r, s, t, u, v, x, y} • Running time for FIND-SET and UNION depends on implementation. • Can be shown to be α(n)=O(logn) where α() is a very slowly growing function.
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